Expositions of topological data analysis traditionally invoke point cloudsdiscrete subsets of some metric space – as the generic mathematical incarnation of datasets to be analyzed.

Maybe a more realistic and more encompassing model for sets of observed data – namely for measurement results – is the time-honored notion of a tuple of (values of) real observables, namely a continuous function

Obs:X n Obs \;\colon\; X \xrightarrow{\phantom{---}} \mathbb{R}^n

to the one-point compactification of nn-dimensional Cartesian space.

A feature in the data as seen by such an observable is then an isosurface of ObsObs, hence the pre-image

Obs 1(v){xX|Obs(x)=v}X Obs^{-1}(\vec v) \;\coloneqq\; \Big\{ x \in X \,\big\vert\, Obs(x) = \vec v \Big\} \;\subset\; X

of a tuple v n\vec v \in \mathbb{R}^n of observed values.

Without restriction of generality we may assume that the observed value of interest is the origin 0 n0 \in \mathbb{R}^n, for if it is instead some v n\vec v \in \mathbb{R}^n then we may instead pass to the observable ObsObsconst vObs \coloneqq Obs - const_{\vec v} without changing the essence of the situation.

In practice, the value of an observable can never be determined with the accuracy of a mathematical point in n\mathbb{R}^n, instead there will be some positive real number r >0r \in \mathbb{R}_{\gt 0} such that one may hope (or wish) to measure ObsObs up to measurement errors within a radius rr. In this case, the desired isosurface could be any element in the set

{f 1(0)X|fC 0(X, n),fObs <r} \Big\{ f^{-1}(0) \subset X \,\vert\, f \in C^0(X,\mathbb{R}^n) ,\, \Vert f - Obs \Vert_\infty \lt r \Big\}

For example, XX might model the interior of a plasma container (say a fusion reactor) and Ob=(T,p):X 2Ob = (T, p) : X \to \mathbb{R}^2 could be the combined temperature- and pressure-observable (say as seen by laser probes into the plasma). Its isosurfaces are the intersections of given isobars with isotherms?.

Illustrating the main theorem of persistent Cohomotopy.

Suppose we want to know the zeros of the observable ff. If the resolution/error bar of measuring ff is rr, then we know that

  1. the zeros must be somewhere away from the gray region.

  2. the homotopy class of ff on the quotient determines the number of zeros mod 2.

Last revised on May 21, 2022 at 16:05:43. See the history of this page for a list of all contributions to it.